On hexagonal gravity water waves

Nicholls, David P.

doi:10.1016/s0378-4754(00)00296-2

Cited by 11 publications

(9 citation statements)

References 35 publications

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“…In contrast with the "diamond-like" waveforms of the solutions above, these surfaces display the "rectangular" or "hexagonal" forms characteristic of higher values of θ (see e.g. Nicholls [16,17], and Craig and Nicholls [18]). Again, in both plots the significant nonlinearity is evident in the long, shallow troughs and sharp, steep crests.…”

Section: Plots Of Prototypical Solutionsmentioning

confidence: 91%

“…These latter authors all derive a nonlinear system of equations from the ideal fluid equations for unknown Fourier coefficients, and then find solutions by Newton iteration and/or numerical continuation. More recent calculations of Nicholls [16,17], and Craig and Nicholls [18] were achieved with a numerical continuation method applied to a Fourier spectral discretization of the Hamiltonian formulation of the water wave equations as posed by Zakharov [19].…”

Section: Introductionmentioning

confidence: 99%

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Stable, high-order computation of traveling water waves in three dimensions

Nicholls

Reitich

2006

European Journal of Mechanics - B/Fluids

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The Euler equations of free-surface ocean dynamics constitute a model of central importance in fluid mechanics due to the wide range of physical phenomena they are intended to represent, from shoaling and breaking of waves in nearshore regions to energy and momentum transport in the open ocean. From a mathematical perspective, these equations present rather unique challenges for analysis and simulation as they couple the subtleties of nonlinear wave equations (balancing nonlinearity with dispersion in the absence of dissipation) to the difficulties of free-boundary problems. In this paper a new, stable high-order boundary perturbation algorithm for the numerical simulation of traveling water waves is described. Its performance is compared to that of classical surface deformation algorithms and it is shown that the new scheme displays significantly enhanced conditioning properties and a lower computational cost, which enable very accurate predictions of physical observables such as velocity, energy, height/steepness, and shape.

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Section: Plots Of Prototypical Solutionsmentioning

confidence: 91%

Section: Introductionmentioning

confidence: 99%

Stable, high-order computation of traveling water waves in three dimensions

Nicholls

Reitich

2006

European Journal of Mechanics - B/Fluids

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“…This set of equations, (2.2), has been useful in a variety of analytical [8,7] and numerical [16,17,9,14] treatments of the Euler equations, and clearly a detailed understanding of the DNO is at the heart of these analyses.…”

Section: Problem Statement and Change Of Variablesmentioning

confidence: 99%

Analyticity of Dirichlet--Neumann Operators on Hölder and Lipschitz Domains

Hu¹,

Nicholls²

2005

SIAM J. Math. Anal.

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Abstract. In this paper we take up the question of analyticity properties of Dirichlet-Neumann operators with respect to boundary deformations. In two separate results, we show that if the deformation is sufficiently small and lies either in the class of C 1+α (any α > 0) or Lipschitz functions, then the Dirichlet-Neumann operator is analytic with respect to this deformation. The proofs of both results utilize the "domain flattening" change of variables recently advocated by Nicholls and Reitich for the stable, high-order numerical simulation of Dirichlet-Neumann operators. We extend their analyticity results through the use of more specialized function spaces, and our new theorems are optimal in terms of boundary regularity. In the case of C 1+α boundary perturbations the underlying field also lies in the Hölder class C 1+α and the theorem follows by appealing to familiar Schauder theory arguments. In contrast, for Lipschitz deformations the field must lie in an L p -based Sobolev space (W 1,p ), so the relevant elliptic estimates come from Sobolev theory. Additionally, in the case of Lipschitz domains, the Dirichlet-Neumann operator must be reformulated weakly in order to accommodate the lack of regularity at the boundary which these Sobolev-class fields possess.

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“…Figure 6 shows the hexagonal or beehive wave pattern captured during the experiment in front of a vertical wall for the case of θ 0 = 30 • . This is typical of the cross sea generated by the oblique interaction of two or more traveling plane waves (see, e.g., Le Mehauté, 1976;Mei, 1983;Nicholls, 2001). Postacchini et al (2014) studied the dynamics of crossing wave trains on a plane slope in shallow waters.…”

Section: Hydraulic Experimentsmentioning

confidence: 99%

Laboratory and numerical experiments on stem waves due to monochromatic waves along a vertical wall

Yoon

Lee

Kim

et al. 2018

Nonlin. Processes Geophys.

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Abstract. In this study, both laboratory and numerical experiments are conducted to investigate stem waves propagating along a vertical wall developed by the incidence of monochromatic waves. The results show the following features: for small-amplitude waves, the wave heights along the wall show a slowly varying undulation. Normalized wave heights perpendicular to the wall show a standing wave pattern. The overall wave pattern in the case of small-amplitude waves shows a typical diffraction pattern around a semiinfinite thin breakwater. As the amplitude of incident waves increases, both the undulation intensity and the asymptotic normalized wave height decrease along the wall. For largeramplitude waves with smaller angle of incidence, the measured data clearly show stem waves. Numerical simulation results are in good agreement with the results of laboratory experiments. The results of present experiments favorably support the existence and the properties of stem waves found by other researchers using numerical simulations. The characteristics of the stem waves generated by the incidence of monochromatic Stokes waves are compared with those of the Mach stem of solitary waves.

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On hexagonal gravity water waves

Cited by 11 publications

References 35 publications

Stable, high-order computation of traveling water waves in three dimensions

Stable, high-order computation of traveling water waves in three dimensions

Analyticity of Dirichlet--Neumann Operators on Hölder and Lipschitz Domains

Laboratory and numerical experiments on stem waves due to monochromatic waves along a vertical wall

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